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We don't know the definite answer with regard to the same but I personally believe he can't and this is my argument

When Vegeta and Goku fought at the end, they both turned Super Saiyan Blue. Considering Vegeta expected Goku to use Ultra Instinct, we don't see him transforming into that form.

Vegeta unlike Goku, doesn't like to fool around with an opponent(Who he knows is strong) and goes all out. He turned SSJB right off the bat against Weakened Golden Freiza, Hit, Goku Black, Jiren, Toppo(First Fight). So it would only make sense for Vegeta to fight at full power right off the bat considering even he knows Ultra Instinct Goku is incredibly strong after witnessing his fight against Jiren.

Ultra Instinct was referred to as Goku's limit breaking power. Super Saiyan Blue Evolution would be Vegeta's limit breaking power(Based on comments made in Episode 123 by the Grand Priest and Universe 7 spectators). Considering Goku needs to be pushed to the edge to activate his Limit Breaking Power, the same would apply to Vegeta. The same also applied to Jiren considering it was only after him getting overwhelmed by Goku, he managed to awaken his hidden power.

Unlike the first time Goku used Ultra Instinct Omen against Jiren, Goku was fully conscious when he turned Mastered Ultra Instinct considering the fact that he was conversing with Jiren during the course of battle. So that puts away the Vegeta was conscious when he transformed and Goku wasn't argument.

A lot of Vegeta fans might argue that Vegeta's transformation is necessary to make him relative to SSJB + Kaioken Goku. However, Kaioken is a technique. It isn't a transformation. Just like Hit's timeskip gave him the edge over Goku, Goku is going to have the edge over Vegeta with Instant transmission and Kaioken

I believe Vegeta's new transformation has the potential to Rival Ultra Instinct Goku in the future. Jiren's Raw power was strong enough to Rival Mastered Ultra Instinct Goku. So it is definitely possible for Vegeta to become stronger with that transformation and Rival Ultra Instinct Goku in terms of power. I believe Vegeta needs to be pushed to the limit for him to release every bit of energy he has(Based on his comments in Episode 123), which allows him to surpass the level of power of Super Saiyan Blue. Hence, I would conclude by saying Vegeta and Goku both have access to only Super Saiyan Blue. However, when pushed to the edge, they have access to their limit breaking powers. Or perhaps with further training, they might be able to tap into that power by will.